Monoidal structures arising from $B_{\mathrm{\infty }}$-algebras with applications to Hopf algebras

We give an explicit construction of monoidal structures on derived categories of right $A_{\mathrm{\infty }}$-modules over an $A_{\mathrm{\infty }}$-algebra $A$ equipped with a $B_{\mathrm{\infty }}$-structure. Given such a $B_{\mathrm{\infty }}$-algebra $A$, we construct an induction functor $$\iota \text{\hspace{0.17em}⁣}:{\mathcal{D}}_{\mathrm{\infty }}^{\mathrm{r}}(A)⟶{\mathcal{D}}_{\mathrm{\infty }}^{\mathrm{b}\mathrm{i}}(A)$$ from right $A_{\mathrm{\infty }}$-modules to $A_{\mathrm{\infty }}$-bimodules and define $$M{⊠}_{A}N=M{}_A\iota (N)\mathrm{.}$$ We prove that $({\mathcal{D}}_{\mathrm{\infty }}^{\mathrm{r}}(A),{⊠}_A,A)$ is a monoidal triangulated category: the unit and associativity constraints are induced by explicit quasi-isomorphisms of $A_{\mathrm{\infty }}$-bimodules, including $$\iota (A)\simeq A\text{and}\iota (M){}_A\iota (N)\simeq \iota (M{⊠}_{A}N)\mathrm{.}$$ We apply this construction to finite-dimensional Hopf algebras $H$, the Yoneda dg algebra $\mathcal{Y}(\mathbb{k},\mathbb{k})$ of the trivial $H$-module carries a natural brace $B_{\mathrm{\infty }}$-structure, and hence its derived category carries the monoidal structure constructed above. We show that the Koszul duality functor $${\mathrm{H}\mathrm{o}\mathrm{m}}_H(\mathcal{Y}(H,\mathbb{k}),-)\text{\hspace{0.17em}⁣}:\mathcal{K}(\mathrm{I}\mathrm{n}\mathrm{j}\text{-}H)⟶\mathcal{D}(\mathcal{Y}(\mathbb{k},\mathbb{k}))$$ is triangulated lax monoidal, and that its restriction to the localizing subcategory generated by the injective resolution $\mathcal{Y}(H,\mathbb{k})$ is a monoidal triangulated equivalence. If $H$ is local, this localizing subcategory is all of $\mathcal{K}(\mathrm{I}\mathrm{n}\mathrm{j}\text{-}H)$. In particular, this gives an alternative, purely algebraic proof of the monoidal equivalence conjectured by Krause and established by Benson–Krause through the classifying space $BG$. Finally, our examples recover the usual tensor product for graded commutative algebras and show that the resulting brace $B_{\mathrm{\infty }}$ and monoidal structures can depend essentially on the chosen Hopf structure.